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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:9"/><i><font color="Green">E10</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:9_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:9_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="compts_1.html#V6" target="_self">compact</a>
 ;<br/><a NAME="E2:9_1_1"/>
 <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E3:9_1_1"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="compts_1.html#V6" target="_self">compact</a>
 <b>by </b><i><a class="txt" href="compts_1.html#E1:9"><i><font color="Green">E10</font></i></a>, <a class="txt" href="compts_1.html#E1:9_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="compts_1.html#T9" target="_self">Th9</a></i>;<br/><b>hence </b><a NAME="E4:9_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font> is <a href="compts_1.html#V2" target="_self">compact</a>
 <b>by </b><i><a class="ref" href="compts_1.html#T10" target="_self">Th10</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E3:9_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font> is <a href="compts_1.html#V2" target="_self">compact</a>
 ;<br/><a NAME="E4:9_1"/>
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>hence </b><a NAME="E5:9_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="compts_1.html#V6" target="_self">compact</a>
 <b>by </b><i><a class="txt" href="compts_1.html#E1:9_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="compts_1.html#T9" target="_self">Th9</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E3:9"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="pre_topc.html#V2">TopSpace-like</a>
 <b>and </b><br/><a NAME="E4:9"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E5:9"/>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="compts_1.html#E4:9"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a> <b>by </b><i><a class="txt" href="compts_1.html#E3:9"><i><font color="Green">E11</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="compts_1.html#E4:9"><i><font color="Green">E12</font></i></a></i>;<br/>
<a NAME="E8:9"/><i><font color="Green">E13</font></i>: 
(  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="compts_1.html#V6" target="_self">compact</a> iff <font color="Maroon">c<sub>3</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font> is <a href="compts_1.html#V2" target="_self">compact</a> )
 
<b>by </b><i><a class="ref" href="compts_1.html#T10" target="_self">Th10</a></i>;<br/>
<b>hence </b><a NAME="E9:9"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="compts_1.html#V6" target="_self">compact</a> implies <font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font> is <a href="compts_1.html#V2" target="_self">compact</a> )
 <b>by </b><i><a class="txt" href="compts_1.html#E1:9"><i><font color="Green">E10</font></i></a>, <a class="ref" href="compts_1.html#T11" target="_self">Th11</a></i>;<br/>

<b>assume </b><a NAME="E10:9"/>
<font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font> is <a href="compts_1.html#V2" target="_self">compact</a>
 ;<br/>

<a NAME="E11:9"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="compts_1.html#V6" target="_self">compact</a>
 
<b>by </b><i><a class="txt" href="compts_1.html#E8:9"><i><font color="Green">E13</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>hence </b><a NAME="E12:9"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="compts_1.html#V6" target="_self">compact</a>
 <b>by </b><i><a class="txt" href="compts_1.html#E1:9"><i><font color="Green">E10</font></i></a>, <a class="ref" href="compts_1.html#T11" target="_self">Th11</a></i>;<br/>


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